A modified splitting method for the cubic nonlinear Schr\"odinger equation

As a classical time-stepping method, it is well-known that the Strang splitting method reaches the first-order accuracy by losing two spatial derivatives. In this paper, we propose a modified splitting method for the 1D cubic nonlinear Schr\"odinger equation: \begin{align*} u^{n+1}=\mathrm{e}^{i\frac\tau2\partial_x^2}{\mathcal N}_\tau \left[\mathrm{e}^{i\frac\tau2\partial_x^2}\big(\Pi_\tau +\mathrm{e}^{-2\pi i\lambda M_0\tau}\Pi^\tau \big)u^n\right], \end{align*} with ${\mathcal N}_t(\phi)=\mathrm{e}^{-i\lambda t|\Pi_\tau\phi|^2}\phi,$ and $M_0$ is the mass of the initial data. Suitably choosing the filters $\Pi_\tau$ and $\Pi^\tau$, it is shown rigorously that it reaches the first-order accuracy by only losing $\frac32$-spatial derivatives. Moreover, if $\gamma\in (0,1)$, the new method presents the convergence rate of $\tau^{\frac{4\gamma}{4+\gamma}}$ in $L^2$-norm for the $H^\gamma$-data; if $\gamma\in [1,2]$, it presents the convergence rate of $\tau^{\frac25(1+\gamma)-}$ in $L^2$-norm for the $H^\gamma$-data. %In particular, the regularity requirement of the initial data for the first-order convergence in $L^2$-norm is only $H^{\frac32+}$. These results are better than the expected ones for the standard (filtered) Strang splitting methods. Moreover, the mass is conserved: $$\frac1{2\pi}\int_{\mathbb T} |u^n(x)|^2\,d x\equiv M_0, \quad n=0,1,\ldots, L . $$ The key idea is based on the observation that the low frequency and high frequency components of solutions are almost separated (up to some smooth components). Then the algorithm is constructed by tracking the solution behavior at the low and high frequency components separately.